= Solution
Take a homogeneous comoving cell of physical <volume> $V_c=a^3V_0$ and total <entropy> $S=sV_c$. At vanishing <chemical potential>, the reversible <first law of thermodynamics> reads
$$
T\,dS=d(\rho V_c)+P\,dV_c=V_c\,d\rho+(\rho+P)dV_c.
$$
For an extensive equilibrium system, the <thermodynamic Euler relation> is $E=TS-PV_c+\mu N$. At $\mu=0$ this gives
$$
\boxed{s=\frac{\rho+P}{T}.}
$$
The supplied pressure identity verifies consistency directly: $dP=(\rho+P)dT/T$ implies
$$
d\left[\frac{(\rho+P)V_c}{T}\right]=\frac{V_c\,d\rho+(\rho+P)dV_c}{T}=dS.
$$
The <thermodynamic Euler relation> fixes the additive normalization that a differential identity alone would leave undetermined.
For adiabatic cosmological expansion, the <cosmological perfect-fluid continuity equation> gives $\dot\rho+3H(\rho+P)=0$, while $\dot V_c=3HV_c$. Hence
$$
T\dot S=V_c\bigl[\dot\rho+3H(\rho+P)\bigr]=0,\qquad
\boxed{sa^3=\mathrm{constant}.}
$$
This proves <cosmological entropy conservation> in the stated perfect-fluid approximation. <Kinetic equilibrium> alone does not guarantee adiabatic evolution: entropy injection, dissipative stresses or out-of-equilibrium reactions would require additional source terms. Those effects are implicitly excluded in this calculation.
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